= Solution
Transform only the time variable. Two <integrations by parts> give $\mathcal F_t[y_{tt}]=(i\omega)^2\widetilde y=-\omega^2\widetilde y$, when the ordinary transform boundary terms vanish; the same derivative rule holds for <tempered distributions>. Spatial derivatives and the fixed spatial delta commute with the transform. Thus
$$
-\omega^2\widetilde y=\widetilde y_{xx}+\widetilde F(\omega)\delta(x-x_0),
$$
or
$$
\boxed{\widetilde y_{xx}=-\omega^2\widetilde y-\widetilde F(\omega)\delta(x-x_0).}
$$
The spatial <Dirichlet boundary conditions> transform to $\widetilde y(\omega,0)=\widetilde y(\omega,L)=0$.
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